Non-Markovian evolution

At the same time, Prigogine and his co-workers14 15,17 developed a general theory of non-equilibrium statistical mechanics. They derived a non-Markovian evolution equation for the velocity distribution function. Their results contain a generalization of the Boltzmann equation for arbitrary concentration and coupling parameter. This generalization is the long-time limit of their evolution equation. 

Abstract Interaction between a quantum system and its surroundings - be it another similar quantum system, a thermal reservoir, or a measurement device - breaks down the standard unitary evolution of the system alone and introduces open quantum system behaviour. Coupling to a fast-relaxing thermal reservoir is known to lead to an exponential decay of the quantum state, a process described by a Lindblad-type master equation. In modern quantum physics, however, near isolation of individual quantum objects, such as qubits, atoms, or ions, sometimes allow them only to interact with a slowly-relaxing near-environment, and the consequent decay of the atomic quantum state may become nonexponential and possibly even nonmonotonic. Here we consider different descriptions of non-Markovian evolutions and also hazards associated with them, as well as some physical situations in which the environment of a quantum system induces non-Markovian phenomena. 

Keywords Irreversible time evolution, Master Equation, non-Markovian evolution... 

The dynamical contents ofEq. (14.89) is muchmore involved than its Markovian counterpart. Indeed, non-Markovian evolution is a manifestation of multidimensional dynamics, since the appearance of a memory kernel in an equation of motion signifies the existence of variables, not considered explicitly, that change on the same timescale. Still, the physical characteristics of the barrier crossing process remain the same, leading to similar modes of behavior ... 

Billinton et al. 1981 Fegan 1984 Nahman and Mijuskovic 1985), while non-Markovian evolutions have also been condsidered (Singh and Reza Ehrahimian 1982 El-Damcese 2009). Petri nets have also been used (Fricks and Trivedi 1997). 

Abstract. This article reviews from both theoretical and numerical aspects three non-equivalent complete second-order formulations of quantum dissipation theory, in which both the reduced dynamics and the initial canonical thermal equilibrium are properly treated in the weak system-bath coupling limit. Two of these formulations are rather familiar as the time-local and the memory-kernel prescriptions, while another which can be termed as correlated driving-dissipation equations of motion will be shown to have the combined merits of the two conventional formulations. By exploiting the exact solutions to the driven Brownian oscillator system, we demonstrate that the time-local and correlated driving-dissipation equations of motion formulations are usually better than their memory-kernel counterparts, in terms of their applicability to a broad range of system-bath coupling, non-Markovian, and temperature parameters. Numerical algorithms are detailed for an efficient evaluation of both the reduced canonical thermal equilibrium state and the non-Markovian evolution at any temperature, in the presence of arbitrary time-dependent external fields. 

The quantum theory of spectral collapse presented in Chapter 4 aims at even lower gas densities where the Stark or Zeeman multiplets of atomic spectra as well as the rotational structure of all the branches of absorption or Raman spectra are well resolved. The evolution of basic ideas of line broadening and interference (spectral exchange) is reviewed. Adiabatic and non-adiabatic spectral broadening are described in the frame of binary non-Markovian theory and compared with the impact approximation. The conditions for spectral collapse and subsequent narrowing of the spectra are analysed for the simplest examples, which model typical situations in atomic and molecular spectroscopy. Special attention is paid to collapse of the isotropic Raman spectrum. Quantum theory, based on first principles, attempts to predict the. /-dependence of the widths of the rotational component as well as the envelope of the unresolved and then collapsed spectrum (Fig. 0.4). 

The second term of Eq. (40) gives the contribution from collisions. These are non-instantaneous processes since the variation of p 0> at the time t depends on the value of this function at the earlier instant t. The evolution is non-Markovian and the system remembers its earlier history. However, this memory extends only over a finite period, as one can see from the expression (44) for the kernel G, (t). This results from supposing that the poles z( are not infinitesimally close to the real axis and thus that the collision time tc is finite (see Eq. (39)). 

The second part (sections H and I) is devoted to a detailed discussion of the dynamics of unimolecular reactions in the presence and the absence of a potential barrier. Section H presents a critical examination of the Kramers approach. It is stressed that the expressions of the reaction rates in the low-, intermediate-, and high-friction limits are subjected to restrictive conditions, namely, the high barrier case and the quasi-stationary regime. The dynamics related to one-dimensional diffusion in a bistable potential is analyzed, and the exactness of the time dependence of the reaction rate is emphasized. The essential results of the non-Markovian theory extending the Kramers conclusions are also discussed. The final section investigates in detail the time evolution of an unimolecular reaction in the absence of a potential barrier. The formal treatment makes evident a two-time-scale description of the dynamics. 

Be aware of the fact that we have to consider the non-Markovian version of the quantum master equation to stay at a level of description where the emission rate, Eq. (39), can be deduced. Moreover, to be ready for a translation to a mixed quantum classical description a variant has been presented where the time evolution operators might be defined by an explicitly time-dependent CC Hamiltonian, i.e. exp(—iHcc[t — / M) has been replaced by the more general expression Ucc(t,F). 

They may be obtained by means of the matrix IET but only together with the kernel E(f) = F(t) specified by its Laplace transformation (3.244), which is concentration-independent. However, from the more general point of view, Eqs. (3.707) are an implementation of the memory function formalism in chemical kinetics. The form of these equations shows the essentially non-Markovian character of the reversible reactions in solution the kernel holds the memory effect, and the convolution integrals entail the prehistoric evolution of the process. In the framework of ordinary chemical kinetics S(/j = d(t), so that the system (3.707) acquires the purely differential form. In fact, this is possible only in the limit when the reaction is entirely under kinetic control. 

In this limit, however, non-Markovian effects are no longer important because co(E - Eg) 0 so that co(E) so the evolution becomes Markovian near the barrier. Eqs. (5.49) and (5.50) then imply that Eq. (5.54) is identical to Eq. (2.41) in this limit. 

Equation (19) describes a non-Markovian process in the CV space. In fact, the forces acting on the CVs depend explicitly on their history. Due to this non-Markovian nature, it is not clear if, and in which sense, the system can reach a stationary state under the action of this dynamics. In [32] we introduced a formalism that allows to map this history-dependent evolution into a Markovian process in the original variable and in an auxiliary field that keeps track of the visited configurations. Defining... 

What is the significance of the Markovian property of a physical process Note that the Newton equations of motion as well as the time-dependent Schrodinger equation are Markovian in the sense that the future evolution of a system described by these equations is fully determined by the present ( initial ) state of the system. Non-Markovian dynamics results from reduction procedures used in order to focus on a relevant subsystem as discussed in Section 7.2, the same procedures that led us to consider stochastic time evolution. To see this consider a universe described by two variables, zi and z, which satisfy the Markovian equations of motion... 

This equation describes the dynamics in the zi subspace, and its non-Markovian nature is evident. Starting at time t, the future evolution of zi... 

Why has the Markovian time evolution (7.50) of a system with two degrees of freedom become a non-Markovian description in the subspace of one of them Equation (7.51) shows that this results from the fact that Z2(t) responds to the historical time evolution of zi, and therefore depends on past values of zi, not only on its value at time t. More generally, consider a system A B made of a part (subsystem) A that is relevant to us as observers, and another part, B, that affects the relevant subsystem through mutual interaction but is otherwise uninteresting. The non-Markovian behavior of the reduced description of the physical subsystem A reflects the fact that at any time t subsystem A interacts with the rest of the total system, that is, with B, whose state is affected by its past interaction with A. In effect, the present state of B carries the memory of past states of the relevant subsystem A. 

In principle, the presence of slow stochastic torques directly affecting the solute reorientational motion can be dealt with in the framework of generalized stochastic Fokker-Planck equations including frequency-dependent frictional terms. However, the non-Markovian nature of the time evolution operator does not allow an easy treatment of this kind of model. Also, it may be difficult to justify the choice of frequency dependent terms on the basis of a sound physical model. One would like to take advantage of some knowledge of the physical system under... 

As p 0", we have v oo, and as p oo, we find that v grows with p. In consequence, a minimum velocity exists, but no maximum velocity. In the previous section we considered the cases when the microscopic transport processes are described by Markovian random walks. The great advantage of the Hamilton-Jacobi formulation of the front propagation problem in general, and the formulas (4.46) in particular, is that they allow us to study quite complicated transport operators for the evolution of the scalar field p and the underlying random walk model, including non-Markovian processes [118,121,125]. 

This equation describes the time evolution of the projected density f (T, t). The first term in (33) evolves in the subspace of the s-system the second term acts as force produced by the evolution in the m-system which has a projection onto the s-system the last term has a non-Markovian character, as can be seen from the memory kernel iPLe QLP. The non-Markovian character of Eq. (33) is a consequence of the procedure of variable contraction. 

Recently we have constructed a complete second-order QDT (CS-QDT), in which all excessive approximations, except that of weak system-bath interaction, are removed [38]. Besides two forms of CS-QDT corresponding to the memory-kernel COP [Eq. (1.2)] and the time-local POP [Eq. (1.3)] formulations, respectively, we have also constructed a novel CS-QDT that is particularly suitable for studying the effects of correlated non-Markovian dissipation and external time-dependent field driving. This paper constitutes a review of the three nonequivalent CS-QDT formulations [38] from both theoretical and numerical aspects. Concrete comparisons will be carried out in connection with the exact results for driven Brownian oscillator systems, so that sensible comments on various forms of CS-QDT can be reached. Note that QDT shall describe not only the evolution of p(t), but also the reduced thermal equilibrium system as p t oo) = peq(7 )-... 

To construct a CS-QDT formulation, one shall treat the effects of system-bath coupling H [Eq. (2.1c)] to the second order exactly for not only the reduced density operator p t) evolution, but also the initial canonical state of the total composite system, Pt( o —oo) = p (T), before the external field excitation. Various CS-QDT formulations differ at their partial resummation schemes for the higher order contributions. We have recently arrived at three forms of CS-QDT in terms of differential equations of motion [38]. Two of them are in principle equivalent to the conventional second-order COP [Eq. (1.2)] and POP [Eq. (1.3)] formulations (cf. AppendixB). For the sake of clarity, we shall present here only the unconventional one that may be particularly suitable for the numerical study of non-Markovian dissipation in the presence of external time-dependent fields. 

The time evolution of Agg(t) [see Eqs. (649a)-(649c)] can be described in terms of a chain of non-Markovian equations of motion... 

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